Chapter 12: Kinematics of a Particle
Particle:
Particle:
•
• A A particle has a mass particle has a mass but negligible size and but negligible size and shape.shape.
•
• A particle can undergo only translational A particle can undergo only translational motion.motion.
•
• Obects li!e roc!ets" proectile" or #ehicle may be considered Obects li!e roc!ets" proectile" or #ehicle may be considered
as particles pro#ided motion of the body is characterized by
as particles pro#ided motion of the body is characterized by
motion of its mass center $a point% and any rotation of the
motion of its mass center $a point% and any rotation of the
body is neglected.
body is neglected.
•
• Chapter 12 & 1' are de#oted to dynamics of particles. Chapter 12 & 1' are de#oted to dynamics of particles.
Rigid Body:
Rigid Body:
A
A rigid body has rigid body has mass" size and shape. Amass" size and shape. A rigid body can rigid body can
undergo both translational and rotational motion.
undergo both translational and rotational motion.
Chapters 1( ) 21 are de#oted to rigid body dynamics.
12.1: Introduction - An Overview of
12.1: Introduction - An Overview of
Mechanics
Mechanics
*tatics:
*tatics: +he study of +he study of
bodies in e,uilibrium.
bodies in e,uilibrium.
Dynamics:
Dynamics:
1.1. ine!aticsine!atics & concerned -ith & concerned -iththe geometric aspects of motion
the geometric aspects of motion
2.
2. ineticsinetics ) concerned -ith ) concerned -ith
the forces causing the motion
the forces causing the motion
echanics:
echanics: +he study of ho- bodies +he study of ho- bodies
react to forces acting on them.
R"#$I%I&"AR I&"MA$I#' ('ection 12.2)
%inear !otion $also called rectilinear motion% is motion along a straight line" and can therefore be described mathematically using only one spatial dimension.
Position of a Particle
+his is step & 1 in particle analysis.
+he position of the particle at any instant" relati#e to the fi/ed
origin" O" is defined by the position #ector r " or the scalar s. *calar s can be positi#e or negati#e. +ypical units for r and s are meters $m% or feet $ft%.
*is+lace!ent of a Particle
0isplacement is a #ector ,uantity.
0isplacement inal Position & 3nitial Position
+he total distance tra#eled by the particle" s+" is a positi#e scalar that represents the total length of the path o#er -hich the particle tra#els.
+he displacement of the particle is defined as the change in its position. 4ector form: r = r’ - r *calar form:
∆
s s5 ) sA particle is mo#ing in a circular path of radius r . +he magnitude of displacement and distance after half a circle -ould be:
$a% 6ero $b% 7 r $c% 2 r $d% 27 r
,"%O#I$
4elocity is a measure of the rate of change in the position of a particle. 3t is a #ector ,uantity $it has both magnitude and direction%.
athematically" # ds8dt
0irection: A positi#e #alue of #elocity indicates that the particle is
mo#ing along the assumed positi#e direction of the coordinate a/is. A negati#e #alue indicates that the particle mo#es in the opposite
direction.
+he magnitude of the #elocity is called speed" -ith units of m8s or ft8s. *peed is a scalar ,uantity.
A##"%"RA$IO&
Acceleration is the rate of change in the #elocity of a particle. 3t is a #ector ,uantity. +ypical units are m8s2 or ft8s2.
As the te/t indicates" the deri#ati#e e,uations for #elocity and acceleration can be manipulated to get a ds # d#
+he instantaneous acceleration is the time deri#ati#e of #elocity.
4ector form: a dv 8 dt
*calar form: a d# 8 dt d2s 8 dt2
Acceleration can be positi#e $speed
increasing% or negati#e $speed decreasing" also called as decelerating%.
'MMAR O/ I&"MA$I# R"%A$IO&': R"#$I%I&"AR MO$IO&
0ifferentiate position to get #elocity and acceleration.
# ds8dt
a d#8dt
9/ample: +he acceleration of a particle is a
function of its displacement and is gi#en as: a
( x$1 ; kx
2%< -here k is a constant. =i#en: #$ x
% >.' ft8s ? #$ x .@'ft% 1'ft8s. ind k .
+hin!: hich one of the follo-ing three e,uation
-ill help you find k .
# ds8dt
a d#8dt
a ds # d#
'P"#IA% #A'": #O&'$A&$ A##"%"RA$IO& MO$IO&
+he three !inematic e,uations can be integrated for the special case -hen acceleration is constant $a ac% to obtain #ery useful e,uations. A common e/ample of constant acceleration is gra#ity< i.e." a body freely falling to-ard earth. 3n this case" ac g B.1 m8s2 D2.2 ft8s2
do-n-ard. +hese e,uations are:
t
a
#
#
c o+
=
yields=
∫
∫
t o c v vdt
a
dv
o 2 c o o s t $182% a t # s s=
+
+
yields=
∫
∫
t o s dt v ds o % s ) $s 2a % $# # o c 2 o 2=
+
yields=
∫
∫
s s c v vo o dsa
dv v+he three !inematic e,uations are:
# #
; a
ct
s s
;#
t ; .' a
ct
2#
2 #
2
; 2a
c$s & s
%
Eemember: +o use the abo#e three e,uations:
1% +he particle must be mo#ing -ith constant acceleration.
2% Fou must define a fi/ed coordinate a/is and specify positi#e8 negati#e directions before you use these e,ns.
D% +he magnitudes and signs of s" # and ac" used in the abo#e three e,uations are determined -.r.t to the chosen coordinate a/is.
9/ample: An airplane begins its ta!e)off run at A
-ith zero #elocity and a constant acceleration
a. Kno-ing that it becomes airborne D s later
at B and that the distance AB is 2> ft"
determine $a% the acceleration a" $b% the ta!e)off
#elocity G B.
9/ample: Automobiles A and B are tra#eling in adacent high-ay lanes and at t ha#e the positions and speeds sho-n. Kno-ing that automobile A has a constant
acceleration of .( m8s2 and that B has a constant
deceleration of .@ m8s2" determine $a% -hen and -here A
-ill o#erta!e B" $b% the speed of each automobile at that time.
MO$IO& O/ A PRO"#$I%" ('ection 12.0) : : : tan
−
→
∑
=
=
⇒
=
⇒
=
⇒
=
x x x x x Horizontal Component F Ma a dv dt v Cons t tan−
↑
∑
=
= −
⇒
= −
⇒
=
y y y y Vertical Component F Ma Mg a g a Cons tConstant acceleration motion along y a/is
3f -e ignore the air drag8 air)friction on the particle" then the only force acting on a proectile is its -eight
Proectile motion can be treated as t-o rectilinear
motions" one in the horizontal direction e/periencing
zero acceleration and the other in the #ertical
direction e/periencing constant acceleration $i.e."
from gra#ity%.
I&"MA$I# "A$IO&': 3ORI4O&$A% MO$IO&
*ince a/ " the #elocity in the horizontal direction remains constant $#/ #o/% and the position in the / direction can be determined by: / /o ; $#o/% t # # ; act s s ;#t ; .' act2 #2 #2 ; 2ac $s & s%
I&"MA$I# "A$IO&': ,"R$I#A% MO$IO&
*ince the positi#e y)a/is is directed up-ard" ay & g.
Application of the constant acceleration e,uations yields: #y #oy & g t y yo ; $#oy% t & H g t2 #y2 # oy2 & 2 g $y & yo% # # ; act s s ;#t ; .' act2 #2 #2 ; 2ac $s & s%
IMPOR$A&$ "A$IO&' /OR PRO"#$I%" MO$IO&
or Iorizontal otion: / / ; $#/% t
or 4ertical otion: #y #y & g t
y y ; $#y% t & H g t2
#y2 #
y2 & 2 g $y & y%
3mportant: +ime" Jt is the only common term that
9/ample: A s!i umper starts -ith a horizontal ta!e)off #elocity of 2' m8s and lands on a straight landing hill inclined at DL. 0etermine $a% the time bet-een ta!e)off and landing" $b% the length d of the ump.
9/ample: A golf ball is struc! -ith a #elocity of ft8s as sho-n. 0etermine the distance" d " to -here it -ill land.
9/ample: A pump is located near the edge of the horizontal platform sho-n. +he nozzle at A discharges -ater -ith an
initial #elocity of 2' ft8s at an angle of ''M -ith the
#ertical. 0etermine the range of #alues of the height h for -hich the -ater enters the opening BC .
12.5: 6"&RA% #R,I%I&"AR MO$IO& - Introduction
+he path of motion of a plane
A roller coaster car
Cur#ilinear motion occurs -hen the particle mo#es along a cur#ed path.
#R,I%I&"AR MO$IO&
*ince the path is often described in D)0" #ector analysis -ill be used to formulate the particle5s position" #elocity and
acceleration.
+he position of the particle at any instant is designated by the #ector
r r $t%. Noth the magnitude and direction of r may #ary -ith time. A particle mo#es along a cur#e
,"%O#I$
4elocity represents the rate of change in the position of a particle.
+he instantaneous #elocity is the time)deri#ati#e of position
v dr 8dt .
+he #elocity #ector" v" is al-ays tangent to the path of motion.
A##"%"RA$IO&
Acceleration represents the rate of change in the #elocity of a particle.
+he instantaneous acceleration is the time)deri#ati#e of #elocity:
A&A%'I' O/ #R,I%I&"AR MO$IO&
1% sing Eectangular $/" y" z% Components
2% sing ormal and +angential Components $n)t %
D% Cylindrical Coordinates $r" Q" z%
#R,I%I&"AR MO$IO&: R"#$A&6%AR #OMPO&"&$' ('ection 12.7)
3t is often con#enient to describe the motion of a particle in
terms of its /" y" z or rectangular components" relati#e to a fi/ed frame of reference.
+he magnitude of the position #ector is: r $/2 ; y2 ; z2%.'
+he position of the particle can be defined at any instant by the
position #ector
r x i ; y j ; z k .
+he /" y" z components may all be functions of time" i.e."
R"#$A&6%AR #OMPO&"&$': ,"%O#I$
+he magnitude of the #elocity #ector is # R$#/%2 ; $# y% 2 ; $# z% 2S.'
+he direction of v is tangent to the path of motion.
+he #elocity #ector is the time deri#ati#e of the position #ector:
v dr 8dt d$/i %8dt ; d$y j %8dt ; d$zk %8dt
*ince the unit #ectors i " j " k are constant in magnitude and direction" this e,uation reduces to # #/i ; #y j ; #zk
R"#$A&6%AR #OMPO&"&$': A##"%"RA$IO&
+he direction of a is usually not tangent to the path of the particle.
+he acceleration #ector is the time deri#ati#e of the
#elocity #ector $second deri#ati#e of the position #ector%:
a dv8dt d2r 8dt2 a / i ; ay j ; azk -here a/ d#/8dt" ay d#y 8dt" az d#z8dt
#
//
#
yy
#
zz
• •• •• •• • •+he magnitude of the acceleration #ector is a R$a/%2 ; $a
y%
2 ; $a z%
"8a!+le: +he bo/ slides do-n the slope described by the e,uation y $.' x2% m" -here x is in meters.
#/ )D m8s" a x )1.' m8s2 at x ' m. ind the y
components of the #elocity and the acceleration of the bo/ at x ' m.
12.9 #R,I%I&"AR MO$IO&
&ORMA% A&* $A&6"&$IA% #OMPO&"&$'
O;ective: 0etermine the #elocity and acceleration of a particle tra#eling along a cur#ed path using normal and tangential components.
&ORMA% A&* $A&6"&$IA% #OMPO&"&$'
$*ection 12.>%
hen a particle mo#es along a cur#ed path" it is sometimes con#enient to describe its motion using coordinates other than Cartesian. hen the path of motion is !no-n" normal $n% and tangential $t % coordinates are
often used.
3n the n-t coordinate system" the origin" O" is located on the
particle $the origin mo#es -ith the particle%.
+he t )a/is is tangent to the path $cur#e% at the instant considered" positi#e in the direction of the particle5s motion.
+he n)a/is is perpendicular to the t )a/is -ith the positi#e direction to-ard the center of cur#ature of the cur#e.
&ORMA% A&* $A&6"&$IA% #OMPO&"&$'
$continued%
+he positi#e n and t directions are defined by the unit #ectors un and ut" respecti#ely.
+he center of cur#ature" OT" al-ays lies on the conca#e side of the cur#e. +he radius of cur#ature"
ρ
" is defined as the perpendicular distance from the cur#e to the center of cur#ature at that point.,"%O#I$ I& $3" n-t #OOR*I&A$" ''$"M
+he #elocity #ector is al-ays tangent to the path of motion $t )direction%.
A##"%"RA$IO& I& $3" n-t #OOR*I&A$" ''$"M
Acceleration is the time rate of change of #elocity :
a dv8dt d$#ut%8dt #. ut ; #u. t
.
a v ut ; $#28
ρ
% un atut ; anun.
After mathematical manipulation" the acceleration #ector can be
A##"%"RA$IO& I& $3" n-t #OOR*I&A$" ''$"M
$continued%
*o" there are t-o components to the acceleration #ector:
a atut ; anun
• +he normal or centripetal component is al-ays directed to-ard the center of cur#ature of the cur#e. an #28
ρ
• +he tangential component is tangent to the cur#e and in the direction of increasing or decreasing #elocity.
at #.
• +he magnitude of the acceleration #ector is a R$at%2 ; $a
n%
3POE+A+ 9UA+3O*
+he magnitude of the acceleration #ector is
a R$a
t%
2; $a
n
%
2S
.'+angential Acceleration" a
t d#8dt
ormal Acceleration" a
n #
28V
'P"#IA% #A'"' O/ MO$IO&
+here are some special cases of motion to consider.
1% +he particle mo#es along a straight line.
ρ
∞
W an #28ρ = 0 =>
a at #
+he tangential component represents the time rate of change in the magnitude of the #elocity.
.
2% +he particle mo#es along a cur#e at constant speed. at # . W a an #28
ρ
'P"#IA% #A'"' O/ MO$IO& $continued%
D% *ometimes" the e,uation of the cur#e path follo-ed
by the particle may be gi#en" y $ x%. +he radius of
cur#ature" V" at any point on the path can be calculated
from the follo-ing e,uation:
D 2 2 2 2 1
dy
dx
d y
dx
ρ
+
÷
=
"8a!+le: A oat travels around a circular +ath at a s+eed that increases with ti!e< v = (>.>027 t 2) !?s. /ind the
!agnitudes of the oat@s velocity and acceleration at the instant t = 1> s.
"8a!+le: $he !otorcyclist travels along the curve at a constant s+eed of > ft?s. *eter!ine his acceleration when he is located at +oint A. 3int: $reat the !otorcycle and rider as a +article.
"8a!+le: $he auto!oile has a s+eed of > ft?s at +oint A and an acceleration< a< having a !agnitude of 1> ft?s2< acting
in the direction shown. *eter!ine the radius of curvature of the +ath at +oint A and the tangential co!+onent of
R"%A$I,"-MO$IO& A&A%'I' O/ $CO PAR$I#%"'
All motion is relati#e to some frame of reference
2 trains approaching each other $along a line% at B' !m8h each. Obser#ers on either train see the other coming at B' ; B' 1B !m8h. Obser#er on ground sees
±
B' !m8h.R"%A
R"%A$$I," I," PO'I$IO&PO'I$IO& $*ection 12.1%$*ection 12.1%
+he
+he absolute positionabsolute position of t-o of t-o
particles A
particles A and N -ith respect toand N -ith respect to
the fi/ed /"
the fi/ed /" y" y" z reference frame arez reference frame are
gi#en by
gi#en by r r A A and and r r B B. +he. +he position of position of
N relati#e to A
N relati#e to A is represented by is represented by
r
r BA BA r r A A r r B B $#ector $#ector addition%addition%
r
r BA BA r r B B & & r r A A
+herefore" if
+herefore" if r r B B $1 $1 i i ; 2 ; 2 j j % m% m
and
and r r A A $@$@ i i ; ' ; ' j j % m"% m"
then
then r r BA BA $( $( i i & D & D j j % m.% m.
ote:
ote: r r B!A B!A means position of means position of B B as obser#ed from as obser#ed from A A
r
R"%A$I," ,"%O#I$
R"%A$I," ,"%O#I$
+o determine the
+o determine the relati#e #elocityrelati#e #elocity of N of N
-ith respect to A" the time deri#ati#e o
-ith respect to A" the time deri#ati#e off
the relati#e position e,uation is ta!en.
the relati#e position e,uation is ta!en.
v
v BA BA vv B B & & vv A A
or
or
v
v B B vv A A ;; vv BA BA
3n these e,uations"
3n these e,uations" v v B B and and vv A A are called are called absolute #elocitiesabsolute #elocities
and
and vv BA BA is the is the relati#e #elocityrelati#e #elocity of N of N -ith respect to A.-ith respect to A.
ote that
R"%A$
R"%A$I," I," A##"%"RA$A##"%"RA$IO&IO&
+he time deri#ati#e of the relati#e
+he time deri#ati#e of the relati#e
#elocity e,uation yields a similar
#elocity e,uation yields a similar
#ector relationship bet-een the
#ector relationship bet-een the
absolute
absolute and and relati#e accelerationsrelati#e accelerations
of particles A and N.
of particles A and N.
+hese deri#ati#es yield:
+hese deri#ati#es yield: aa BA BA aa B B & & aa A A
or
or
a
'O%,I&6 PROB%"M'
*ince the relati#e motion e,uations are
#ector e,uations" problems in#ol#ing
them re,uires resol#ing #elocity $or
acceleration% #ector along x and y and
then performing the #ector operation:
v
B
v
A;
v
BAa
B
a
A;
a
BA"DAMP%": 'hown in figure are two air+lanes flying at sa!e altitude with their res+ective velocities and direction shown. /ind vB?A.
6iven: #A (' !m8h #N !m8h
"DAMP%" (continued%
'olution:
v A $(' i % !m8h
v B & cos ( i & sin ( j
$ &@ i & (B2. j % !m8h
v BA v B & v A $&1' i & (B2. j % !m8h
# B 8 A
= (−1050)
2+(−692.8)
2= 1258
!m8hθ tan)1$ % DD.@
°
θ
1050
692.8
9/ample: At the instant sho-n" race car A is passing race car B -ith a relati#e #elocity $#A8N% of 1 m8s. Kno-ing that the speeds of both cars are constant and that the relati#e
acceleration of car A -ith respect to car B $aA8N% is .2' m8s2
directed to-ard the center of cur#ature" determine the speed of cars A and B at the instant sho-n.
"8a!+le: At a given instant in an air+lane race< air+lane A is flying horiEontally in a straight line< and its s+eed is eing increased at a rate of 0 !?s2. Air+lane B is flying at the sa!e altitude as
air+lane A and is following a circular +ath of 2>>-! radius. nowing that at the given instant the s+eed of B is eing decreased at the rate of 2 !?s2< deter!ine< for the +ositions
AB'O%$" *"P"&*"&$ MO$IO& A&A%'I' O/ $CO PAR$I#%"' ('ection-12.F)
3n many !inematics
problems" the motion of one obect -ill depend on the motion of another
obect.
• 3f bloc! A mo#es do-n-ard along the inclined plane" bloc! N -ill mo#e up the other incline.
• +his dependency commonly occurs if the particles are interconnected by ine/tensible8 inelastic cords -hich are -rapped around pulley$s%.
PRO#"*R" /OR A&A%'I'- *"P"&*"&$ MO$IO&
+he motion of each bloc! can be related mathematically by
defining position coordinates" sA and sN. 9ach coordinate a/is is defined from a fi/ed point or datum line" measured positi#e along each plane in the direction of motion of each bloc!.
ote: +he length of the cord connecting the t-o particles -ill al-ays remain same8 constant" irrespecti#e of the position8 location of the t-o particles.
*"P"&*"&$ MO$IO&
$continued%
3n this e/ample" position
coordinates sA and sN can be defined from fi/ed datum lines e/tending from the center of the pulley along each incline to bloc!s A and N.
*ince" the cord has a fi/ed length" the position coordinates sA and sN are related mathematically by the e,uation
sA ; lC0 ; sN l+
Iere l+ is the total cord length and lC0 is the length of cord passing o#er the arc C0 on the pulley.
*"P"&*"&$ MO$IO&
$continued%
+he negati#e sign indicates that as A mo#es do-n the incline $positi#e sA direction%" N mo#es up the incline $negati#e sN direction%.
Accelerations can be found by differentiating the #elocity e/pression. aN )aA.
dsA8dt ; dsN8dt W #N )#A
+he #elocities of bloc!s A and N can be related by differentiating the position e,uation. ote that lC0 and l+ remain constant" so dlC08dt dl+8dt
*"P"&*"&$ MO$IO&: PRO#"*R"'
+hese procedures can be used to relate the dependent motion of particles mo#ing along rectilinear paths $only the magnitudes of
#elocity and acceleration change" not their line of direction%.
@. 0ifferentiate the position coordinate e,uation$s% to relate #elocities and accelerations. Keep trac! of signsX
D. 3f a system contains more than one cord" relate the position of a point on one cord to a point on another
cord. *eparate e,uations are -ritten for each cord. 2. Eelate the position coordinates to the cord length.
*egments of cord that do not change in length during the motion may be left out.
1. 0efine position coordinates from fi/ed datum lines"
along the path of each particle. 0ifferent datum lines can be used for each particle.
"DAMP%"-1
=i#en: 3n the figure on the left" the cord at A is pulled do-n -ith a speed of 2 m8s. ind: +he speed of bloc! N.
